Determinants are kind of abstract and even inelegant. This is because it's defined in terms of coordinates rather than abstractly. A coordinate free definition exists, but it's very complicated.
Ideally they would not be taught at all, but it's useful for doing exam problems. Cramer's rule in particular is quite useful when solving simultaneous equations in engineering under exam conditions, for example. Even in computer/engineering applications, row reduction/diagonal normal form type are used instead.
Determinants are a way to get a scalar number out of a matrix. You have linear forms, bilinear forms, and multilinear forms, the latter of which determinants fall under.
There are certain properties you would like the determinant to satisfy, such as det(AB) = det(A) det(B), so that the determinant can give you an idea of whether a matrix is invertible. Eventually you work out that a determinant _has to be_ a multilinear form with certain properties (alternating, skew symmetric).
Again this is all very complex and messy. Initially 'linear algebra' was just the study of determinants, because people thought of math in terms of solving equations. Linear algebra, as it's known today, came later.
Bilinear forms turn up all the time, so ideally you would first learn bilinear forms, then learn the properties a bilinear form can have (e.g. alternating, skew symmetry, etc), and then learn determinants in that context.